How do you rewrite the expression as a simplified expression containing one term?

#cos (alpha + beta) cos beta + sin (alpha + beta) sin beta#

1 Answer

#cos(alpha+beta)cosbeta+sin(alpha+beta)sinbeta=cosalpha#

Explanation:

#cos(alpha+beta)cosbeta+sin(alpha+beta)sinbeta#

Remember that:

#sin(a+b)=sinacosb+cosasinb#

and

#cos(a+b)=cosacosb-sinasinb#

#:. (cosalphacosbeta-sinalphasinbeta)cosbeta+(sinalphacosbeta+cosalphasinbeta)sinbeta#

#cosalphacos^2beta-sinalphasinbetacosbeta+sinalphacosbetasinbeta+cosalphasin^2beta#

#cosalphacos^2beta+cosalphasin^2beta#

#cosalpha(cos^2beta+sin^2beta)#

Remember that #sin^2x+cos^2x=1#, so:

#cosalpha(1)=cosalpha#